Extensions of the Wiener Number

Abstract

Particularly for structure−property correlations there are many chemical graph-theoretic indices, one of which is Wiener's “path number”. Because Wiener's original work focused on acyclic structures, one can imagine different ways of extending it to cycle-containing structures, several of which are noted here. Many of these different formulas in fact yield like numerical values for general connected graphsthat is, different formulas sometimes correspond to the same graph invariant. Indeed it is found that there are two “dominant” classes of formulas each corresponding to one of two distinct invariants. Extensions to sequences of invariants (with the Wiener index the first member) are more often found to give distinct sequences. Further a powerful vector-space theoretic view for characterizing and for comparing different sequences is described. This is illustrated for a collection of eight sequences in application to a set of molecular graph structures (corresponding to the octanes). Another type of extension is to generate a sequence or partially ordered set of graph invariants for which the Wiener index is the natural first member of this set. Certain such sets of invariants (corresponding to contributions from different types of subgraphs) are noted to be “complete” (or form a basis) in the sense that any invariant can be faithfully linearly expanded in terms of the members of the set.

Maximum topological distances based indices as molecular descriptors for QSPR

“…Results show that resorting to the detour matrix for defining the topological indices yields better correlations to predict enthalpies of formation. These results agree with other similar ones to study other physical chemistry properties, which seems to support the use of detour indices in structure-property modeling [25][26][27][61][62][63]. We conclude that the obtained results are good enough for the chosen set to validly infer that the detour matrix ∆ represents a convenient topological device to be employed in the QSAR/QSPR analysis and it constitutes a valuable molecular descriptor which verily adds to the set of topological matrices.…”

Some inequalities for the Kirchhoff index of graphs

“…Gutman and Mohar [12] (see also [13]) proved that the Kirchhoff index can be obtained from the non-zero eigenvalues of the Laplacian matrix:…”

Relations Between Resistance Distances of a Graph and its Complement or its Contraction

“…While various invariants (such as the analogue to the Wiener index-the Kirchhoff index, 1 the degree Kirchhoff index 9 and the additive degree Kirchhoff index 10 ) may be used as another one of the many topological indices to make QSPRs or QSARs, [11][12][13][14][15][16][17][18] we believe that the resistance distance has a more fundamental role. Being a natural intrinsic metric for graphs, which indeed is motivatable in different ways, 9,[19][20][21][22][23] such as then indicates is diverse relevance.…”